What is the Islamic legal ruling for a man who commits adultery with a married woman and fathers a child by her, and is his repentance valid, and how does he atone for his sin, and what is the fate of the child resulting from this relationship?

The child born of adultery is not attributed to the adulterer if the woman is married; rather, he is attributed to her husband, because the Prophet, peace and blessings be upon him, said: "The child is for the bed, and for the adulterer is the stone." So the husband is the one to whom the child is ascribed in lineage and inheritance, unless he disavows him through li'an. As for the adulterer, he has no right whatsoever to the child.